Sum & Difference Formulas for Tangent

tan(α + β) = tanα + tanβ/1 – tanαtanβ – Sum

tan(α - β) = tanα - tanβ/1 + tanαtanβ – Difference

Ex 1. Suppose tanα = 1/3 and tanβ = ½

a. Find tan(α + β)

tanα + tanβ/1 – tanαtanβ = 1/3 + ½ /1 – (1/3)(1/2)

2/6 + 3/6 / 1 – 1/6 = 5/6 / 5/6 = 1

b. Find tan(α – β)

tanα - tanβ/1 + tanαtanβ = 1/3 – ½ /1 + (1/3)(1/2)

2/6 – 3/6 / 1 + 1/6 = -1/6 / 7/6 = -6/42 = -1/7

Ex 2. tanα = 2/3 and tanβ = ½

a. tanα + tanβ/1 – tanαtanβ = 2/3 + ½ /1 – (2/3)(1/2)

4/6 + 3/6 / 1 – 2/6 = 7/6 / 2/3 = 21/12 = 7/4

b. tanα - tanβ/1 + tanαtanβ = 2/3 – ½ /1 + (2/3)(1/2)

4/6 – 3/6 / 1 + 2/6 = 1/6 / 4/3 = 3/24 = 1/8